Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets

A Survey of Interval‐Valued Fuzzy Sets

“…The algorithm proposed by Tizhoosh for images with L levels of gray consists of the following steps: We must point out that Tizhoosh calls E F (see again [61]) ultrafuzziness and that the algorithm above has been generalized in [51,52]. In Figure 22.2 we give an example where it is proved that we obtain the best result with this algorithm.…”

“…The first, E F , was presented in 1996 in [61], and the second, E c , in 2001 in [91]. The difference between the two definitions lies in the fact that E F is a measure of how far an IVFS is from a…”

“…Lemma 1, together with other properties (see [21,49,54,61]) of the operators K α , has made it possible to prove the following characterization theorem (which is an adaptation for IVFSs of Theorem 3.6 proved in [77]). …”

“…In [61], these entropies E F are studied in depth and a theorem for their construction from functions ϕ : [0, 1] → [0, 1] is presented. It is also said there that the most commonly used expression is the one obtained when we take ϕ(x) = x so that E F (A) = J (A) for all A ∈ IVFSs(U ).…”

“…Historically, the expressions developed in Definition 7 were defined without including the term relative to the length of the intervals (see [4,34,60,61]). …”